Test 3 Questions

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Test 3 Questions Chapter 10 10.1 Let S=$100, K=$105, r=8%, T =0.5, and δ=0 .Let u =1.3, d=0.8, and n= 1. a. What are the premium, ∆ , and B for a European call? b. What are the premium, ∆ , and B for a European put? 10.2 Let S=$100, K=$95, r=8%, T =0.5, and δ=0. Let u=1.3, d=0.8, and n=1. a. Verify that the price of a European call is $16.196. b. Suppose you observe a call price of $17. What is the arbitrage? c. Suppose you observe a call price of $15.50. What is the arbitrage? 10.3 Let S=$100, K=$95, r=8%, T =0.5, and δ=0. Let u=1.3, d=0.8, and n=1. a. Verify that the price of a European put is $7.471. b. Suppose you observe a put price of $8. What is the arbitrage? c. Suppose you observe a put price of $6. What is the arbitrage? 10.4 Obtain at least 5 years’ worth of daily or weekly stock price data for a stock of your choice. a. Compute annual volatility using all the data. b. Compute annual volatility for each calendar year in your data. How does volatility vary over time? c. Compute annual volatility for the first and second half of each year in your data. How much variation is there in your estimate? 10.5 Obtain at least 5 years of daily data for at least three stocks and, if you can, one currency. Estimate annual volatility for each year for each asset in your data. What do you observe about the pattern of historical volatility over time? Does historical volatility move in tandem for different assets? 10.6 Let S=$100, K=$95, σ=30%, r=8%, T=1, and δ=0. Let u=1.3, d=0.8, and n = 2. Construct the binomial tree for a call option. At each node provide the premium, ∆ , and B.

10.7 Repeat the option price calculation in the previous question for stock prices of $80, $90, $110, $120, and $130, keeping everything else fixed. What happens to the initial option ∆ as the stock price increases? 10.8 Let S=$100, K=$95, σ=30%, r=8%, T=1,and δ=0. Let u=1.3, d=0.8, and n = 2. Construct the binomial tree for a European put option. At each node provide the premium, ∆ , and B. 10.9 Repeat the option price calculation in the previous question for stock prices of $80, $90, $110, $120, and $130, keeping everything else fixed. What happens to the initial put ∆ as the stock price increases? 10.10 Let S=$100, K=$95, σ=30%, r=8%, T=1, and δ=0. Let u=1.3, d=0.8, and n = 2. Construct the binomial tree for an American put option. At each node provide the premium, ∆ , and B. 10.11 Suppose S 0 = $100, K = $50, r = 7.696% (continuously compounded), δ = 0, and T = 1. a. Suppose that for h = 1, we have u = 1.2 and d = 1.05. What is the binomial option price for a call option that lives one period? Is there any problem with having d > 1? b. Suppose now that u = 1.4 and d = 0.6. Before computing the option price, what is your guess about how it will change from your previous answer? Does it change? How do you account for the result? Interpret your answer using put-call parity. c. Now let u = 1.4 and d = 0.4. How do you think the call option price will change from (a)? How does it change? How do you account for this? Use put-call parity to explain your answer. 10.12 Let S = $100, K = $95, r = 8% (continuously compounded), σ = 30%, δ = 0, T =1 year, and n=3. a. Verify that the binomial option price for an American call option is $18.283. Verify that there is never early exercise; hence, a European call would have the same price. b. Show that the binomial option price for a European put option is $5.979. Verify that put-call parity is satisfied.

c. Verify that the price of an American put is $6.678. 10.13 Repeat the previous problem assuming that the stock pays a continuous dividend of 8% per year (continuously compounded). Calculate the prices of the American and European puts and calls. Which options are early-exercised? 10.14 Let S=$40, K=$40, r=8% (continuously compounded), σ=30%, δ=0, T= 0.5 year, and n = 2. a. Construct the binomial tree for the stock. What are u and d? b. Show that the call price is $4.110. c. Compute the prices of American and European puts. 10.15 Use the same data as in the previous problem, only suppose that the call price is $5 instead of $4.110. a. At time 0, assume you write the option and form the replicating portfolio to offset the written option. What is the replicating portfolio and what are the net cash flows from selling the overpriced call and buying the synthetic equivalent? b. What are the cash flows in the next binomial period (3 months later) if the call at that time is fairly priced and you liquidate the position? What would you do if the option continues to be overpriced the next period? c. What would you do if the option is underpriced the next period? 10.16 Suppose that the exchange rate is $0.92/ € . Let r $ = 4%, and r € = 3%, u = 1.2, d=0.9, T =0.75, n=3, and K=$0.85. a. What is the price of a 9-month European call? b. What is the price of a 9-month American call? 10.17 Use the same inputs as in the previous problem, except that K = $1.00. a. What is the price of a 9-month European put? b. What is the price of a 9-month American put? 10.18 Suppose that the exchange rate is 1 dollar for 120 yen. The dollar interest rate is 5% (continuously compounded) and the yen rate is 1% (continuously

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compounded). Consider an at-the-money American dollar call that is yen- denominated (i.e., the call permits you to buy 1 dollar for 120 yen). The option has 1 year to expiration and the exchange rate volatility is 10%. Let n = 3. a. What is the price of a European call? An American call? b. What is the price of a European put? An American put? c. How do you account for the pattern of early exercise across the two options? 10.19 An option has a gold futures contract as the underlying asset. The current 1- year gold futures price is $300/oz, the strike price is $290, the risk-free rate is 6%, volatility is 10%, and time to expiration is 1 year. Suppose n = 1. What is the price of a call option on gold? What is the replicating portfolio for the call option? Evaluate the statement: “Replicating a call option always entails borrowing to buy the underlying asset.” 10.20 Suppose the S&P 500 futures price is 1000, σ=30%, r=5%, δ=5%, T=1, and n = 3. a. What are the prices of European calls and puts for K = $1000? Why do you find the prices to be equal? b. What are the prices of American calls and puts for K = $1000? c. What are the time-0 replicating portfolios for the European call and put? 10.21 For a stock index, S=$100, σ=30%, r=5%, δ=3%, and T=3. Let n=3. a. What is the price of a European call option with a strike of $95? b. What is the price of a European put option with a strike of $95? c. Now let S = $95, K = $100, σ = 30%, r = 3%, and δ = 5%. (You have exchanged values for the stock price and strike price and for the interest rate and dividend yield.) Value both options again. What do you notice? 10.22 Repeat the previous problem calculating prices for American options instead of European. What happens? 10.23 Suppose that u < e (r−δ)h . Show that there is an arbitrage opportunity. Now suppose that d > e (r−δ)h . Show again that there is an arbitrage opportunity.

Examples from text Figure 10.4 Binomial tree for pricing a European call option; assumes S = $41.00, K = $40.00, σ = 0.30, r = 0.08, T = 2.00 years, δ = 0.00, and h = 1.000. At each node the stock price, option price, ∆ , and B are given. Option prices in bold italic signify that exercise is optimal at that node. Figure 10.10 Binomial tree for pricing an American call option on a futures contract; assumes S = $300.00, K = $300.00, σ = 0.10, r = 0.05, T = 1.00 years, δ = 0.05, and h = 0.333. At each node the stock price, option price, ∆ , and B are given. Option prices in bold italic signify that exercise is optimal at that node.

Figure 10.9 Binomial tree for pricing an American put option on a currency; assumes S = $1.05 / € , K = $1.10, σ = 0.10, r = 0.055, T = 0.50 years, δ = 0.031, and h = 0.167. At each node the stock price, option price, ∆ , and B are given. Option prices in bold italic signify that exercise is optimal at that node. Figure 10.8 Binomial tree for pricing an American call option on a stock index; assumes S = $110.00, K = $100.00, σ = 0.30, r = 0.05, T = 1.00 years, δ = 0.035, and h = 0.333. At each node the stock price, option price, ∆ , and B are given. Option prices in bold italic signify that exercise is optimal at that node.

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Figure 10.7 Binomial tree for pricing an American put option; assumes S = $41.00, K = $40.00, σ = 0.30, r = 0.08, T = 1.00 years, δ = 0.00, and h = 0.333. At each node the stock price, option price, ∆ , and B are given. Option prices in bold italic signify that exercise is optimal at that node. Chapter 11

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